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Magnetism
Magnets and Magnetic Fields
We studied electric forces and electric fields yesterday…
Now we study magnetism and magnetic fields.
Rather than defining magnetism, we begin by discussing
properties of magnetic materials.
Recall how there are two kinds of charge (+ and -), and likes
repel, opposites attract.
Similarly, there are two kinds of magnetic poles (North and
South), and like poles repel, opposites attract.*
S
S
N
S
N
Repel
S
N
N
S
N
Thanks to
Dr. Waddill
for the nice
pictures!
S
N
N
S
N
Repel
S
Attract
Attract
*Recall also that I have a mental defect which often causes me
to say “likes attract and unlikes repel” when I mean the
opposite. I am not to be penalized for a mental defect!
There is one difference between magnetism and electricity:
it is possible to have isolated + or – electric charges, but
isolated N and S poles have never been observed.
-
+
S
N
I.E., every magnet has BOTH a N and a S pole, no how
many times you “chop it up.”
S
N
=
S
N
+ S
N
The earth has associated with it a magnetic field, with poles
near the geographic poles.
The pole of a magnet attracted to
the earth’s north geographic pole
is the magnet’s North pole.
N
The pole of a magnet attracted
to the earth’s south geographic
pole is the magnet’s South pole.
S
http://hyperphysics.phyastr.gsu.edu/hbase/magnetic/
magearth.html
Just as we used the electric field to help us “explain” and
visualize electric forces in space, we use the magnetic field
to help us “explain” and visualize magnetic forces in
space.
Magnetic field lines point in the same direction that the
north pole of a compass would point.
Magnetic field lines are tangent to the magnetic field.
The more magnetic field lines in a region in space, the
stronger the magnetic field.
Outside the magnet, magnetic field lines point away from N
poles (*why?).
Huh?
Nooooooo….
*The N pole of a compass would “want to get to” the S pole of the magnet.
Is the earth’s north pole a magnetic N or a magnetic S?
It has to be a S, otherwise, the
compass N would not point to it.
Unless the N of a compass
needle is really S. Dang! This is
too much for me!
Yup, it’s confusing.
Here’s a “picture” of the magnetic
field of a bar magnet, using iron
filings to map out the field.
The magnetic field ought to
“remind” you of the earth’s field.
Later I’ll give a better definition for magnetic
field direction.
Here’s what the magnetic field looks like when you put
unlike or like poles next to each other.
The magnetic field B is a vector which points in the
direction of magnetic field lines. We will quantify the
magnitude of B later.
Electric Current Produces Magnetism
An electric current produces a magnetic field.*
The direction of the current is given by
the right-hand rule. Grasp the currentcarrying wire in your right hand, with
your thumb pointing in the direction of
the current. Curl your fingers around
the wire. Your fingers indicate the
direction of the magnetic field.
*Experimentally observed, then demanded by theory as a logical
consequence of Maxwell’s equations.
Picture on previous page is from
http://physics.mtsu.edu/~phys232/Lectures/
L12-L16/L17/Current_Loops/current_loops.html
This picture also illustrates the magnetic field due to a
current-carrying loop of wire.
These symbols mean
“out of page” and “into
page.” See next
section.
Field comes out
of page here.
“Turns around” and
goes into page here.
Here’s a simpler case: the magnetic field due to a straight
wire.
Field comes out
of page here.
Field turns
around and
goes into page
here.
Thanks again
to Dr. Waddill
for the nice
pictures!
If the wire is grasped in the right hand with the thumb
pointing in the direction of the current, the fingers will curl
in the direction of B.
Force on an Electric Current in a Magnetic Field;
Definition of B
As seen above, an electric current gives rise to a magnetic
field, which must exert a force on a magnet.
Does a magnet exert a force on a current-carrying wire?
(Newton’s 3rd Law says it should.)
Yes—a current-carrying wire in a magnetic field “feels” a
force. The direction is given by the right-hand rule:
Point your outstretched fingers in the direction of the
current. Bend your fingers 90º and orient your hand to
point the bent fingers in the direction of the magnetic field.
Your thumb points in the direction of the force.
You may need to re-orient your hand as you go through this
procedure. During exams, I see all sorts of gyrations as
students try to figure out directions.
I’ll demonstrate another right-hand rule in class.
Here is a web “physics toy” to help you visualize the force
on a current-carrying conductor. (Select Lorentz Force.)
Below is another picture to help you visualize. It came from
http://hyperphysics.phy-astr.gsu.edu/hbase/magnetic/forwir2.html.
The web page even
has a built-in
calculator that gives
you numerical
answers to your force
problems!
The force is perpendicular to both the current and the
magnetic field.
You would expect the magnitude of the force to depend on
the magnitudes of the magnetic field and the current. In
fact, it does.
The force also depends on how much of the wire is in the
magnetic field.
If the direction of the current is perpendicular to the
magnetic field, then F = I ℓ B. I’ll write the lowercase l in
italics (ℓ) to help you distinguish it from the number 1.
If the current and magnetic field are not perpendicular, the
force is given by
F = I ℓ B sin ,
OSE:
where  is the angle between the current vector and the
magnetic field vector. (Smallest angle from the current
vector to the magnetic field vector.)
I
ℓ

B
The right-hand rule and the equation above actually serve
as the definition of the magnetic field B.
The SI unit for magnetic field is the tesla:
1 T = 1 N / (1 A · 1 m).
A 1 tesla magnetic field is extremely strong. The earth’s
magnetic field is a few hundredths of a tesla.
I have to go to a lot of effort to explain magnetic field and
force direction when I teach the non-calculus course.
It’s so much easier with calculus and vectors. The force on
a charge q moving with a velocity v in a magnetic field B is
found to obey F = qv  B.
The magnitude of the cross product is qvB sin . But it’s
so much easier learning the right-hand rule for the vector
cross product, and applying it to torques, charged
particles, etc., instead of learning a seemingly new right
hand rule for each new topic. The elegance of math!
If you take a number of charged particles in a volume of
wire that has a length ℓ in a magnetic field, it is easy to
derive the vector form of our OSE: F = I  B.
Example In the figure two slides back, B=0.9 T, I=30 A, ℓ=12
cm, and =60°. What is the force on the wire?
F = I ℓ B sin  = (30 A) (0.12 m) (0.9 T) (sin 60°)
F = 2.8 N
I
ℓ

B
Hold it! Force is a vector quantity! What is the direction.
The force is perpendicular to both current direction and
magnetic field direction. Apply either version of the righthand rule and you find it is into the paper.
You could use the right-hand screw rule, which is the way I
best visualize the direction. Or just let the math tell you!
We need to have a way to draw 3-d vectors on 2-d paper.
We will use the symbol
for a vector pointing directly out
of the page, towards us (that is supposed to look like the
sharp point of an arrowhead coming right towards your
eye).
We will use the symbol  for a vector pointing directly into
the page, away from us (that is supposed to look like the
feathered end of an arrow going away from your eye).
Example A rectangular loop of
wire hangs vertically in a
magnetic field B as shown. B is
uniform along the 10 cm
horizontal length of wire, and the
top portion of the wire is outside
the field. The loop hangs from a
balance which measures a
downward force F=3.48x10-2 N in
excess of the wire weight when
the current is 0.245 A. What is
the magnitude of B?
Huh?
I
I
10 cm
       
       
       
B
       
F
Sorry, that’s not an acceptable answer. You do know how
to work this problem!
The forces on these vertical
segments of the wire are equal and
opposite in direction. We need not
worry about them further.
I
I
10 cm
I can figure out the directions of
those two forces. Can you?
The force on the lower horizontal
segment is downward, as shown
in the drawing. Could you verify
that?
The angle between current and
magnetic field is 90°. sin(90°)=1.
OSE:
F = I ℓ B sin 
       
       
       
B
       
F
solve for B!
B = F  ( I ℓ sin(90) )
B = (3.48x10-2 N)  (0.245 A) (0.1 m)
B = 1.42 T
Hey, that wasn’t so bad! Only the direction bit is hard work
at this point.
Force on an Electric Charge Moving in a Magnetic
Field
We’ve kind of done this already—what’s the difference
between a moving charge and a current in a wire?
The current was confined to a wire, but we don’t expect
that to alter the forces involved.
The algebra-based textbook does thing backwards from
normal—force on wire first, then force on charge.
I already gave the equation for the force on a charge:
F = qv  B
F = qvB sin 
You can see the “derivation” in lec27_long.ppt,
supplementary material.
Here’s the equation we use if we “can’t” use cross
products:
OSE :
F = q v B sin θ .
If the charged particle is moving perpendicular to B,  = 90°
and the force is greatest: F = q v B.
The above OSE gives the magnitude of the force. The right
hand rule gives the direction for positive charges.
For negative charges, just reverse the direction (determine
the direction as if it were for a positive charge and the force
on the negative charge is in the opposite direction).
         
Bout
v         
v
 +
     
-
FB
FB
         
         
         
Thanks again to Dr. Waddill for the nice picture. Don’t you
wish you were taking Physics 24 too?
Magnetic Field due to a Straight Wire
We already saw how the
magnetic field due to a current
“curls around” a wire. This tells
us the direction of the magnetic
field. What about the
magnitude?
Experimentally it is found (and verified by theory) that the
larger the current, the larger the magnetic field, and the
further away from the wire, the weaker the magnetic field.
Mathematically,
OSE :
μ0 I
B=
,
2π r
where I is the current in the wire, r is the distance away
from the wire at which B is being measured, and 0 is a
constant:
Tm
μ0 = 4π  10-7
.
A
This “funny” definition of 0 allows us to more elegantly
define current (later).
Example A vertical electric wire in the wall of a building
carries a current of 25 A upward. What is the magnetic field
at a point 10 cm due north of this wire?
Let’s make north be to the left in this
picture, and up be up.
According to the right hand rule, the N
magnetic field is to the west, coming
out of the plane of the “paper.”
To calculate the magnitude, B:
OSE :
B=
μ0 I
,
2π r

-7 T m 
 4π  10
  25 A 
A 
B= 
= 5  10-5 T .
2π
 0.1 m 
up
I=25 A
B
d=0.1 m
Definition of the Ampere and the Coulomb
We defined the ampere of current this morning as being 1 C
of charge flowing past a point in 1 s: 1 A = 1 C / 1 s.
That’s the way I learned it many years ago.
Now the ampere is actually defined as the current flowing
in two parallel wires 1 m apart which produces a force per
unit length of 2x10-7 N/m.
A coulomb is then defined as 1 A · 1 s.
Physics is constantly being “tweaked” as new knowledge
and experimental techniques become available.
Ampere’s Law
What is a solenoid?
A solenoid is a coil of wire with many
loops.
Each loop produces a magnetic field
that looks like this.
“When the coils of the solenoid are closely spaced, each
turn can be regarded as a circular loop, and the net
magnetic field is the vector sum of the magnetic field for
each loop. This produces a magnetic field that is
approximately constant inside the solenoid, and nearly
zero outside the solenoid.”
Thanks again to Dr. Waddill for the pictures and text.
“The ideal solenoid is approached when the coils are very
close and the length of the solenoid is much greater than
its radius. Then we can approximate the magnetic field as
constant inside and zero outside the solenoid.”
B
Textbooks show that the magnetic field inside the
solenoid is
B = μ0 n I .

The slice is made perpendicular to
the wires and parallel to the solenoid
axis.
I

The vectors in and out of the page
represent the current (and therefore
the wires), so imagine this picture as
a slice through the center of the
solenoid, perpendicular to the wires.
B = μ0 n I
B is the magnitude of the magnetic field inside the solenoid
(the direction is given by the right-hand rule), n is the
number of loops per unit length (loops per meter), and I is
the current in the wire.
I’ll write the “official” version like this:
OSE :
B = μ0
N
I
L
N is the total number of loops (sometimes called “turns”)
and L is the total length of the solenoid.
More about solenoids on-line here.
The magnetic field of a solenoid looks like the magnetic
field of a bar magnet.
(http://hyperphysics.phy-astr.gsu.edu/hbase/magnetic/elemag.html#c1)
The BIG IDEAS
There are two BIG IDEA equations buried in this lecture.
It is not obvious where they are, because we are so
focused on details when we learn this material for the first
time.
One of the big ideas arises from the observation that
magnetic poles always come in pairs, unlike + and –
charged particles.
In the next lecture, I’ll introduce the idea of magnetic flux,
which is “like” the idea of electric flux.
You can calculate the magnetic flux through a given area:
If you integrate the magnetic flux over a closed area (e.g.,
a sphere, or a cylinder closed at both ends), the result is
zero:
The integral is zero because wherever you find a N pole,
you also find a S pole, and the net flux going out of the
surface must equal the net flux going into the surface
(kind of like the N “cancels” the S).
The equation is called Gauss’ law for magnetism, and is
one of Maxwell’s four equations.
It also says there is no such thing as a magnetic
monopole. Some quantum theories suggest that
magnetic monopoles might exist. We have not found
them. If we do, then the right hand side of the equation
above will need modified.
You also saw Ampere’s law, which appeared in the
context of a solenoid. The law is far more general than
that. It also appeared in the equation for a magnetic field
due to a current in a wire, except then we didn’t call it
Ampere’s law.
OSE :
μ0 I
B=
2π r
OSE :
B = μ0
N
I
L
If you integrate this expression over a closed path, you
get a result proportional to the current I and the total path
length.
In the next lecture, we will find that electric fields which
change with time also give rise to magnetic fields, so the
full version of the Ampere’s law Maxwell equation is
You’ve seen three out of Maxwell’s four equations. One
more lecture on E&M, one more equation!